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We show that it is possible to consistently describe dynamical systems, whoseequations of motion are of degree higher than two, in the microcanonicalensemble, even if the higher derivatives aren't coordinate artifacts. Highertime derivatives imply that there are more than one Hamiltonians, conservedquantities due to time translation invariance, and, if the volume in phasespace, defined by their intersection, is compact, microcanonical averages canbe defined and there isn't any instability, in the sense of Ostrogradsky, eventhough each Hamiltonian, individually, may define a non-compact (hyper)surface.  We provide as concrete example of these statements the Pais--Uhlenbeckoscillator and show that it can describe a system that makes sense in themicrocanonical ensemble. It describes two oscillators that are coupled byimposing a fixed phase difference, that thereby describes a non--localinteraction between them. The consistent quantum dynamics can straightforwardlybe expressed using two pairs of creation and annihilation operators, with thephase difference describing a flux, that describes the interaction.  The properties of the action imply that particular solutions, that woulddescribe independent oscillators, are, in general, not admissible.The reason isthat the coordinate transformation, that would decouple the oscillators isn't asymmetry of the action--unless a "BPS bound" is saturated. Only then do theydecouple. But, in these cases, the action does describe one, not two,oscillators, anyway and the higher derivative term is a coordinate artifact.

Higher time derivatives in the microcanonical ensemble describe dynamics of flux-coupled classical and quantum oscillators